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This paper explores the role of generalized continuum mechanics, and the feasibility of model-free data-driven computing approaches thereof, in solids undergoing failure by strain localization. Specifically, we set forth a methodology for capturing material instabilities using data-driven mechanics without prior information regarding the failure mode. We show numerically that, in problems involving strain localization, the standard data-driven framework for Cauchy/Boltzmann continua fails to capture the length scale of the material, as expected. We address this shortcoming by formulating a generalized data-driven framework for micromorphic continua that effectively captures both stiffness and length-scale information, as encoded in the material data, in a model-free manner. These properties are exhibited systematically in a one-dimensional softening bar problem and further verified through selected plane-strain problems.

This paper presents a multi-temporal formulation for simulating elastoplastic solids under cyclic loading. We leverage the proper generalized decomposition (PGD) to decompose the displacements into multiple time scales, separating the spatial and intra-cyclic dependence from the inter-cyclic variation. In contrast with the standard incremental approach, which solves the (non-linear and computationally intensive) mechanical balance equations at every time step, the proposed PGD approach allows the mechanical balance equations to be solved exclusively for the small-time intra-cyclic response, while the large-time inter-cyclic response is described by simple scalar algebraic equations. Numerical simulations exhibiting complex cyclic responses, including a 2D problem and an application to a monopile foundation, demonstrate that PGD solutions with a limited number of space-time degrees of freedom may be obtained numerically, only requiring a few modes to accurately capture the reference response., Comment: Version accepted for publication in Computer Methods in Applied Mechanics and Engineering

The complexity of many problems in computational mechanics calls for reliable programming codes and accurate simulation systems. Typically, simulation responses strongly depend on material and model parameters, where one distinguishes between backward and forward models. Providing reliable information for the material/model parameters, enables us to calibrate the forward model (e.g., a system of PDEs). Markov chain Monte Carlo methods are efficient computational techniques to estimate the posterior density of the parameters. In the present study, we employ Bayesian inversion for several mechanical problems and study its applicability to enhance the model accuracy. Seven different boundary value problems in coupled multi-field (and multi-physics) systems are presented. To provide a comprehensive study, both rate-dependent and rate-independent equations are considered. Moreover, open source codes (https://doi.org/10.5281/zenodo.6451942) are provided, constituting a convenient platform for future developments for, e.g., multi-field coupled problems. The developed package is written in MATLAB and provides useful information about mechanical model problems and the backward Bayesian inversion setting. © 2022, The Author(s).

The prediction of crack initiation and propagation in ductile failure processes are challenging tasks for the design and fabrication of metallic materials and structures on a large scale. Numerical aspects of ductile failure dictate a sub-optimal calibration of plasticity- and fracture-related parameters for a large number of material properties. These parameters enter the system of partial differential equations as a forward model. Thus, an accurate estimation of the material parameters enables the precise determination of the material response in different stages, particularly for the post-yielding regime, where crack initiation and propagation take place. In this work, we develop a Bayesian inversion framework for ductile fracture to provide accurate knowledge regarding the effective mechanical parameters. To this end, synthetic and experimental observations are used to estimate the posterior density of the unknowns. To model the ductile failure behavior of solid materials, we rely on the phase-field approach to fracture, for which we present a unified formulation that allows recovering different models on a variational basis. In the variational framework, incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. The overall formulation is revisited and extended to the case of anisotropic ductile fracture. Three different models are subsequently recovered by certain choices of parameters and constitutive functions, which are later assessed through Bayesian inversion techniques. A step-wise Bayesian inversion method is proposed to determine the posterior density of the material unknowns for a ductile phase-field fracture process. To estimate the posterior density function of ductile material parameters, three common Markov chain Monte Carlo (MCMC) techniques are employed: (i) the Metropolis–Hastings algorithm, (ii) delayed-rejection adaptive Metropolis, and (iii) ensemble Kalman filter combin

Modeling complex material failure with competing mechanisms is a difficult task that often leads to mathematical and numerical challenges. This work contributes to the study of localized failure mechanisms by means of phase fields in a variational framework: in addition to the treatment of brittle and ductile fracture, done in previous work, we consider the case of shear band formation followed by ductile fracture. To achieve this, a new degradation function is introduced, which distinguishes between two successive failure mechanisms: (i) plastic strain localization and (ii) ductile fracture. Specifically, the onset of elastic damage is delayed to allow for the formation of shear bands driven by plastic deformations, thus accounting for the mechanisms that precede the coalescence of voids and microcracks into macroscopic ductile fractures. Once a critical degradation value has been reached, a phase-field model is introduced to capture the (regularized) kinematics of macroscopic cracks. To tackle the issue of potentially high computational cost, we propose a parallel implementation of the phase-field approach based on an iterative algorithm. The algorithm was implemented within the Alya system, a high performance computational mechanics code. Several examples show the capabilities of our implementation. We pay special attention to the ability to capture different failure mechanisms

This paper presents a framework for modeling failure in quasi-brittle geomaterials under different loading conditions. A micromechanics-based model is proposed in which the field variables are linked to physical mechanisms at the microcrack level: damage is related to the growth of microcracks, while plasticity is related to the frictional sliding of closed microcracks. Consequently, the hardening/softening functions and parameters entering the free energy follow from the definition of a single degradation function and the elastic material properties. The evolution of opening microcracks in tension leads to brittle behavior and mode I fracture, while the evolution of closed microcracks under frictional sliding in compression/shear leads to ductile behavior and mode II fracture. Frictional sliding is endowed with a non-associative law, a crucial aspect of the model that considers the effect of dilation and allows for realistic material responses with non-vanishing frictional energy dissipation. Despite the non-associative law, a variationally consistent formulation is presented using notions of energy balance and stability, following the energetic formulation for rate-independent systems. The material response of the model is first described, followed by the numerical implementation procedure and several benchmark finite element simulations. The results highlight the ability of the model to describe tensile, shear, and mixed-mode fracture, as well as responses with brittle-to-ductile transition. A key result is that, by virtue of the micromechanical arguments, realistic failure modes can be captured, without resorting to the usual heuristic modifications considered in the phase-field literature. The numerical results are thoroughly discussed with reference to previous numerical studies, experimental evidence, and analytical fracture criteria., Comment: Postprint version. Minor corrections

The prediction of crack initiation and propagation in ductile failure processes are challenging tasks for the design and fabrication of metallic materials and structures on a large scale. Numerical aspects of ductile failure dictate a sub-optimal calibration of plasticity- and fracture-related parameters for a large number of material properties. These parameters enter the system of partial differential equations as a forward model. In this work, we develop a step-wise Bayesian inversion framework for ductile fracture to provide accurate knowledge regarding the effective mechanical parameters. To this end, synthetic and experimental observations are used to estimate the posterior density of the unknowns. To model the ductile failure behavior of solid materials, we rely on the phase-field approach to fracture, for which we present a unified formulation that allows recovering different models on a variational basis. In the variational framework, incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. The overall formulation is revisited and extended to the case of anisotropic ductile fracture. Three different models are subsequently recovered by certain choices of parameters and constitutive functions, which are later assessed through Bayesian inversion techniques. To estimate the posterior density function of ductile material parameters, three common Markov chain Monte Carlo (MCMC) techniques are employed: (i) the Metropolis-Hastings algorithm, (ii) delayed-rejection adaptive Metropolis, and (iii) ensemble Kalman filter combined with MCMC. To examine the computational efficiency of the MCMC methods, we employ the R-convergence tool., Comment: 61 pages, 21 figures, 10 tables

The present study was conducted in the Machángara Alto and Chulco rivers, which belong to the Paute basin in the provinces of Azuay and Cañar in southern Ecuador. Study focus: Andean watersheds are important providers of water supply for human consumption, food supply, energy generation, industrial water use, and ecosystem services and functions for many cities in Ecuador and in the rest of South America. In these regions, accurate quantification and prediction of water flow is challenging, mainly due to significant climatic variability and sparse monitoring networks. In the context of flow forecasting, this work evaluates the accuracy of two physical models (WEAP and GR2M) and two models based on artificial neural networks (ANN) that use meteorological data as input variables. Then, a hybrid technique is proposed, using the time series generated by the individual models as inputs of a new ANN. This approach aims to increase the accuracy of the simulated flow by combining and exploiting the information provided by physical and data-driven models. To assess the performance of the proposed methodology, statistical analyses are conducted for two case studies in the Andean region, where comparative analyses are performed for the individual models and the hybrid technique. New hydrological insights: The results indicate that the proposed technique is able to improve the individual performance of physical and ANN-based models, yielding good results in the calibration and validation stages for the two case studies. Specifically, increases in NSE were observed from 0.64 to 0.99 in the MachÁngara Alto river, and from 0.88 to 0.99 in the Chulco river. Higher accuracy of the hybrid technique was observed for all evaluation criteria considered in the analyses. The performance of the hybrid technique was also reflected in terms of water supply and demand, suggesting possible applications for the regional management of water resources, where accurate flow predictions are of utmos

The governing equations of the variational approach to brittle and ductile fracture emerge from the minimization of a non-convex energy functional subject to irreversibility constraints. This results in a multifield problem governed by a mechanical balance equation and evolution equations for the internal variables. While the balance equation is subject to kinematic admissibility of the displacement field, the evolution equations for the internal variables are subject to irreversibility conditions, and take the form of variational inequalities, which are typically solved in a relaxed or penalized way that can lead to deviations of the actual solution. This paper presents an interior-point method that allows to rigorously solve the system of variational inequalities. With this method, a sequence of perturbed constraints is considered, which, in the limit, recovers the original constrained problem. As such, no penalty parameters or modifications of the governing equations are involved. The interior-point method is applied in both a staggered and a monolithic scheme for both brittle and ductile fracture models. In order to stabilize the monolithic scheme, a perturbation is applied to the Hessian matrix of the energy functional. The presented algorithms are applied to three benchmark problems and compared to conventional methods, where irreversibility of the crack phase-field is imposed using a history field or an augmented Lagrangian., Comment: 36 pages, 16 figures

Strain localization is one of the most challenging phenomena in solid mechanics. It occurs when strains concentrate in very narrow bands within a solid, typically referred to as localization bands. This behaviour is related to different types of failure mechanisms: fracture, shear bands and slip lines. Being a dissipative process, the modelling of strain localization could seem to preclude the use of variational methods. However, a sound framework to deal with this issue has been developed in the last few decades. In this contribution, we review the modelling of strain localization by means of variational methods systematically, presenting the main underlying theoretical concepts. The issue of irreversibility is approached by means of the theory of generalized standard materials, which constitutes the basis for the variational approach. Then, typical problems occurring in the modelling of strain localization are analyzed: the tendency to localize in a band of zero thickness with no dissipation, the determination of the geometry of the localization band, and the orientation bias of such band with respect to mesh alignment in finite element discretizations. We discuss solutions for these problems, focusing on the approach that tackles the description of the localization band using phase fields. © 2020, CIMNE, Barcelona, Spain.

[Abstract] Study region The present study was conducted in the Machángara Alto and Chulco rivers, which belong to the Paute basin in the provinces of Azuay and Cañar in southern Ecuador. Study focus Andean watersheds are important providers of water supply for human consumption, food supply, energy generation, industrial water use, and ecosystem services and functions for many cities in Ecuador and in the rest of South America. In these regions, accurate quantification and prediction of water flow is challenging, mainly due to significant climatic variability and sparse monitoring networks. In the context of flow forecasting, this work evaluates the accuracy of two physical models (WEAP and GR2M) and two models based on artificial neural networks (ANN) that use meteorological data as input variables. Then, a hybrid technique is proposed, using the time series generated by the individual models as inputs of a new ANN. This approach aims to increase the accuracy of the simulated flow by combining and exploiting the information provided by physical and data-driven models. To assess the performance of the proposed methodology, statistical analyses are conducted for two case studies in the Andean region, where comparative analyses are performed for the individual models and the hybrid technique. New hydrological insights The results indicate that the proposed technique is able to improve the individual performance of physical and ANN-based models, yielding good results in the calibration and validation stages for the two case studies. Specifically, increases in NSE were observed from 0.64 to 0.99 in the MachÁngara Alto river, and from 0.88 to 0.99 in the Chulco river. Higher accuracy of the hybrid technique was observed for all evaluation criteria considered in the analyses. The performance of the hybrid technique was also reflected in terms of water supply and demand, suggesting possible applications for the regional management of water resources, where accurate flow pr

This paper addresses the modeling of fracture in quasi-brittle materials using a phase-field approach to the description of crack topology. Within the computational mechanics community, several studies have treated the issue of modeling fracture using phase fields. Most of these studies have used an approach that implies the lack of a damage threshold. We herein explore an alternative model that includes a damage threshold and study how it compares with the most popular approach. The formulation is systematically explained within a rigorous variational framework. Subsequently, we present the corresponding three-dimensional finite element discretization that leads to a straightforward numerical implementation. Benchmark simulations in two dimensions and three dimensions are then presented. The results show that while an elastic stage and a damage threshold are ensured by the present model, good agreement with the results reported in the literature can be obtained, where such features are generally absent.

This paper presents a modeling framework to describe the driving mechanisms of cyclic failure in brittle and ductile materials, including cyclic plasticity and fatigue crack growth. A variational model is devised using the energetic formulation for rate-independent systems, coupling a phase-field description of fatigue fracture to a cyclic plasticity model that includes multi-surface kinematic hardening, gradient-enhanced isotropic hardening/softening and ratcheting. The coupled model embeds two distinctive fatigue effects. The first captures the characteristic features of low-cycle fatigue, driven by the accumulation of plastic strains, while the second accounts for high-cycle fatigue, driven by free energy accumulation. The interplay between these mechanisms allows to describe a wide range of cyclic responses under both force loading and displacement loading, as shown in several numerical simulations. Moreover, the phase-field approach to fracture accounts for the initiation and propagation of fatigue-induced cracks., Comment: Minor errata corrections (pages 17 and 18). 48 pages, 22 figures

This paper addresses the modeling of fracture in quasi-brittle materials using a phase-field approach to the description of crack topology. Within the computational mechanics community, several studies have treated the issue of modeling fracture using phase fields. Most of these studies have used an approach that implies the lack of a damage threshold. We herein explore an alternative model that includes a damage threshold and study how it compares with the most popular approach. The formulation is systematically explained within a rigorous variational framework. Subsequently, we present the corresponding three-dimensional finite element discretization that leads to a straightforward numerical implementation. Benchmark simulations in two dimensions and three dimensions are then presented. The results show that while an elastic stage and a damage threshold are ensured by the present model, good agreement with the results reported in the literature can be obtained, where such features are generally absent., Comment: 25 pages, 8 figures

This paper addresses the modeling of fracture in quasi-brittle materials using a phase-field approach to the description of crack topology. Within the computational mechanics community, several studies have treated the issue of modeling fracture using phase fields. Most of these studies have used an approach that implies the lack of a damage threshold. We herein explore an alternative model that includes a damage threshold and study how it compares with the most popular approach. The formulation is systematically explained within a rigorous variational framework. Subsequently, we present the corresponding three-dimensional finite element discretization that leads to a straightforward numerical implementation. Benchmark simulations in two dimensions and three dimensions are then presented. The results show that while an elastic stage and a damage threshold are ensured by the present model, good agreement with the results reported in the literature can be obtained, where such features are generally absent.

High resolution images of environmental variables are highly valuable sources of information in research and environmental management. Obtaining spatially continuous information from ground observations is challenging due to the wide variety of factors that affect classical interpolation methods. While geostatistical methods have produced noteworthy results, they generally rely on important assumptions and strongly depend on the availability of observed data. In turn, satellite‐based or model‐based gridded images generally represent the global spatial structure of environmental variables, but are subject to bias. With the objective of exploiting the benefits of both sources of information, we propose a new mathematical formulation to merge observed data with gridded images of environmental variables using partial differential equations in a variational setting. With a …